ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu
GENERALIZED PICONE’S IDENTITY AND ITS APPLICATIONS
KAUSHIK BAL
Abstract. In this article we give a generalized version of Picone’s identity in a nonlinear setting for thep-Laplace operator. As applications we give a Sturmian Comparison principle and a Liouville type theorem. We also study a related singular elliptic system.
1. Introduction
The classical Picone’s identity states that, for differentiable functionsv >0 and u≥0, we have
|∇u|2+u2
v2|∇v|2−2u
v∇u∇v=|∇u|2− ∇(u2
v )∇v≥0 (1.1)
Later Allegreto-Huang [1] presented a Picone’s identity for the p-Laplacian, which is an extension of (1.1). As an immediate consequence, they obtained a wide array of applications including the simplicity of the eigenvalues, Sturmian comparison principles, oscillation theorems and Hardy inequalities to name a few.
This work motivated a lot of generalization of the Picone’s identity in different cases see [3, 6, 7] and the reference therein. In a recent paper Tyagi [7] proved a generalized version of Picone’s identity in the nonlinear framework, asking the question about the Picone’s identity which can deal with problems of the type:
−∆u=a(x)f(u) in Ω, u= 0 on∂Ω.
where Ω is a open, bounded subset ofRn.
They proved that for differentiable functionsv >0 andu≥0 we have
|∇u|2+|∇u|2
f0(v) + (up
f0(v)∇v
f(v) − ∇u
pf0(v))2=|∇u|2− ∇( u2
f(v))· ∇v≥0 (1.2) wheref(y)6= 0 andf0(y)≥1 for ally6= 0;f(0) = 0.
Moreover|∇u|2− ∇(u2/f(v))· ∇v= 0 holds if and only ifu=cvfor an arbitrary constant c. In this article, we generalize the main result of Tyagi [7] for the p- laplacian operator; i.e, we will give a nonlinear analogue of the Picone’s identity for thep-Laplacian operator.
In this work, we assume the following hypothesis:
2000Mathematics Subject Classification. 35J20, 35J65, 35J70.
Key words and phrases. Quasilinear elliptic equation; Picone’s identity; comparison theorem.
c
2013 Texas State University - San Marcos.
Submitted July 24, 2013. Published November 8, 2013.
1
• Ω denotes any domain inRn.
• 1< p <∞.
• f : (0,∞)→(0,∞) be aC1 function.
2. Main Results
We first start with the Picone’s identity forp-Laplacian.
Theorem 2.1. Let v >0 andu≥0 be two non-constant differentiable functions inΩ. Also assume thatf0(y)≥(p−1)[f(y)p−2p−1] for ally. Define
L(u, v) =|∇u|p−pup−1∇u|∇v|p−2∇v
f(v) +upf0(v)|∇v|p [f(v)]2 . R(u, v) =|∇u|p− ∇( up
f(v))|∇v|p−2∇v.
ThenL(u, v) =R(u, v)≥0. MoreoverL(u, v) = 0a.e. inΩif and only if∇(uv) = 0 a.e. inΩ.
Remark 2.2. Whenp = 2 and f(y) = y we get the Classical Picone’s Identity (1.1) for Laplacian and whenp= 2 we get back its nonlinear version (1.2).
Proof of Theorem 2.1. ExpandingR(u, v) by direct calculation we get L(u, v). To showL(u, v)≥0 we proceed as follows,
L(u, v) =|∇u|p−pup−1∇u|∇v|p−2∇v
f(v) +upf0(v)|∇v|p [f(v)]2
=|∇u|p+upf0(v)|∇v|p
[f(v)]2 −pup−1|∇u||∇v|p−1 f(v) +pup−1|∇v|p−2
f(v) {|∇u||∇v| − ∇u∇v}
=p|∇u|p
p +(u|∇v|)(p−1)q q[f(v)]q
−p q
(u|∇v|)(p−1)q
[f(v)]q −pup−1|∇u||∇v|p−1 f(v) +upf0(v)|∇v|p
[f(v)]2 +pup−1|∇v|p−2
f(v) {|∇u||∇v| − ∇u.∇v}
Recall from Young’s inequality, for non-negativeaandb, we have ab≤ap
p +bq
q (2.1)
where 1p+1q = 1. Equality holds ifap=bq. So using Young’s Inequality we have,
p|∇u|p
p +(u|∇v|)(p−1)q q[f(v)]q
≥pup−1|∇u||∇v|p−1
f(v) (2.2)
Which is possible since bothuandf are non negative. Equality holds when
|∇u|= u
[f(v)]qp|∇v| (2.3)
Again using the fact that,f0(y)≥(p−1)[f(y)p−2p−1] we have upf0(v)|∇v|p
[f(v)]2 ≥ p q
(u|∇v|)(p−1)q
[f(v)]q (2.4)
Equality holds when
f0(y) = (p−1)[f(y)p−2p−1]. (2.5) Combining (2.2) and (2.4) we obtainL(u, v)≥0. Equality holds when (2.3) and (2.5) together with|∇u||∇v|=∇u.∇v holds simultaneously.
Solving for (2.5) one obtainsf(v) =vp−1. So when,L(u, v)(x0) = 0 andu(x0)6=
0, then (2.2) together withf(v) =vp−1 and|∇u||∇v|=∇u.∇v yields,
∇ u v
(x0) = 0.
Ifu(x0) = 0, then∇u= 0 a.e. on{u(x) = 0}and∇ uv
(x0) = 0.
3. Applications
We begin this section with the application of the above Picone’s identity in the nonlinear framework. As is well understood today that Picone’s identity plays a significant role in the proof of Sturmian comparison theorems, Hardy-Sobolev inequalities, eigenvalue problems, determining Morse index etc. In this section, following the spirit of [1], we will give some applications of the nonlinear Picone’s identity.
Hardy type result. We start this part with a theorem which can be applied to prove Hardy type inequality following the same method as in [1].
Theorem 3.1. Assume that there is a v∈C1 satisfying
−∆pv≥λgf(v) v >0 in Ω.
for someλ >0 and nonnegative continuous functiong. Then for anyu∈Cc∞(Ω);
u≥0 it holds that
Z
Ω
|∇u|p≥λ Z
Ω
g|u|p (3.1)
where, f satisfiesf0(y)≥(p−1)[f(y)p−2p−1].
Proof. Let Ω0⊂Ω, Ω0 be compact. Takeφ∈C0∞(Ω),φ >0. By Theorem 2.1, we have
0≤ Z
Ω0
L(φ, v)≤ Z
Ω
L(φ, v)
= Z
Ω
R(φ, v) = Z
Ω
|∇φ|p− ∇( φp
f(v))|∇v|p−2∇v
= Z
Ω
|∇φ|p+∇( φp f(v))∆pv
≤ Z
Ω
|∇φ|p−λ Z
Ω
gφp.
Lettingφ→u, we have (3.1).
Sturmium Comparison Principle. Comparison principles always played an im- portant role in the qualitative study of partial differential equation. We present here a nonlinear version of the Sturmium comparison principle.
Theorem 3.2. Let f1 andf2 are the two weight functions such thatf1< f2 and f satisfiesf0(y)≥(p−1)[f(y)p−2p−1]. If there is a positive solutionusatisfying
−∆pu=f1(x)|u|p−2uforx∈Ω, u= 0 on∂Ω.
Then any nontrivial solution v of
−∆pv=f2(x)f(v) forx∈Ω,
u= 0 on ∂Ω, (3.2)
must change sign.
Proof. Let us assume that there exists a solution v > 0 of (3.2) in Ω. Then by Picone’s identity we have
0≤ Z
Ω
L(u, v) = Z
Ω
R(u, v)
= Z
Ω
|∇u|p− ∇( up
f(v))|∇v|p−2∇v
= Z
Ω
f1(x)up−f2(x)up
= Z
Ω
(f1−f2)up<0,
which is a contradiction. Hence,vchanges sign in Ω.
Liouville type result. In this section we present a Liouville type result for p- Laplacian. Existence of solution for some equation having non-variational struc- ture is generally obtained using the bifurcation method and by obtaining a priori estimates. With this in mind we give a proof of Liouville type result motivated by [5].
Theorem 3.3. Letc0>0,p >1 andf satisfyf0(y)≥(p−1)[f(y)p−2p−1]. Then the inequality
−∆pv≥c0f(v) (3.3)
has no positive solution in Wloc1,p(Rn).
Proof. We start by assuming thatv is a positive solution of (3.3). Choose R >0 and letφ1be the first eigenfunction corresponding to the first eigenvalueλ1(BR(y)) such thatλ1(BR(y))< c0.
Taking φ
p 1
f(v)as a test function, which is valid since by Vazquez maximum principle [8], f(v)φp1 ∈W1,p(BR(y)). Hence,
c0 Z
BR(y)
φp1− Z
BR(y)
|∇φ1|p≤ − Z
BR(y)
R(φ1, v)≤0. Tt follows that
c0≤ R
BR(y)|∇φ1|p R
BR(y)φp1 =λ1(BR(y))< c0,
which is a contradiction.
Quasilinear system with singular nonlinearity. In this part we will start with a singular system of elliptic equations often occurring in chemical heterogeneous catalyst dynamics. We will show that Picone’s Identity yields a linear relationship betweenuandv. For more information on the singular elliptic equations we refer to [2, 4] and the reference therein.
Consider the singular system of elliptic equations
−∆pu=f(v) in Ω
−∆pv=[f(v)]2 up−1 in Ω u >0, v >0 in Ω u= 0, v= 0 on∂Ω.
(3.4)
wheref satisfiesf0(y)≥(p−1)[f(y)p−2p−1]. We have the following result.
Theorem 3.4. Let (u, v) be a weak solution of (3.4)and f satisfy f0(y) ≥(p− 1)[f(y)p−2p−1]. Thenu=c1v wherec1 is a constant.
Proof. Let (u, v) be the weak solution of (3.4). Now for anyφ1andφ2inW01,p(Ω), we have
Z
Ω
|∇u|p−2|∇u|∇φ1dx= Z
Ω
f(v)φ1dx, (3.5)
Z
Ω
|∇u|p−2|∇u|∇φ2dx= Z
Ω
[f(v)]2
up−1 φ2dx. (3.6)
Choosingφ1=uandφ2=up/f(v) in (3.5) and (3.6) we obtain Z
Ω
|∇u|pdx= Z
Ω
uf(v)dx= Z
Ω
∇( up
f(v))|∇v|p−2∇vdx.
Hence we have Z
Ω
R(u, v)dx= Z
Ω
|∇u|p− ∇( up
f(v))|∇v|p−2∇v dx= 0.
By the positivity ofR(u, v) we have thatR(u, v) = 0 and hence
∇(uv) = 0
which givesu=c1v wherec1 is a constant.
Acknowledgements. The author would like to thank the anonymous referee for his/her useful comments and suggestions.
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Kaushik Bal
School of Mathematical Sciences, National Institute for Science Education and Re- search, Institute of Physics Campus, Bhubaneshwar-751005, Odisha, India
E-mail address:[email protected]
